A wetting and drying treatment for the Runge–Kutta discontinuous Galerkin solution to the shallow water equations

Adaptive Modelling of Coupled Hydrological Processes with Application in Water Management

“…The wetting-drying treatment is based on the thin water layer approach of [6]. Simplifying the stability condition in [6], we use the "reflection numerical flux" to prohibit mass transfer through an element boundary as soon as we detect an emptying of this element.…”

“…Simplifying the stability condition in [6], we use the "reflection numerical flux" to prohibit mass transfer through an element boundary as soon as we detect an emptying of this element. After the positivity of the mean water depth in each element is guaranteed, the so-called positive depth operator [6] ensures the positivity of water depth node-wise. See [8] for details of our implementation.…”

Locally Limited and Fully Conserved RKDG2 Shallow Water Solutions with Wetting and Drying

“…In the last decade, the Godunov-type methods incorporated with a Runge-Kutta (RK) discontinuous Galerkin (DG) local discretization [15,16] [10,24]. However, allowing violation of momentum conservation is not theoretically recommended [48], and in practice, this issue is noticed to influence the accuracy and/or the well-balanced property of an RKDG scheme [10].…”

“…However, allowing violation of momentum conservation is not theoretically recommended [48], and in practice, this issue is noticed to influence the accuracy and/or the well-balanced property of an RKDG scheme [10]. Furthermore, among these wetting and drying algorithms [7,10,17,24], no effort has been made to provide stable and accurate discretization to the friction source terms as in the FV methods [11,12,35,38].…”

“…In relation to wetting and drying over irregular topographies, it is often said that a slope limiter may induce drastic numerical instabilities [10,17]. But no factual insights are yet available.…”